A line between a satellite and the earth center of mass is typically called the nadir, while a line between the satellite and the sun is called a sunline. In a satellite programmed for sun-nadir steering these two lines are used as references to control the position of the satellite. In sun-nadir steering the spacecraft yaw axis is oriented toward the earth, generally coincident with the nadir. Any axis in the spacecraft can be pointed in any direction by rotating the spacecraft through two angles about any two fixed spacecraft axes. In conventional sun-nadir steering, the yaw axis is maintained earth pointed and the two rotation axes are chosen as yaw and pitch.
Conventional sun-nadir steering may be explained very simply as follows: 1) the spacecraft is yawed until the sun comes into the roll-yaw plane; 2) the solar array is pointed at the sun by rotating the solar array about a pitch gimbal until the solar array is normal to the sun. A more detailed description of sun-nadir steering may be found in Effects of Solar Radiation Pressure on Satellite Attitude Control by R. J. McElvain, published in Progress in Astronautics and Rocketry, Volume 8, Guidance and Control, published by Academic Press, 1962. The McElvain reference gives the body and wing steering equations, cited therein as equations 19 and 20.
The term sun-nadir steering may be used broadly, and may encompass steering laws that follow conventional sun-nadir steering as described above with substantive yawing and array pitch rotation over a significant period of time. Some examples include sun-target steering, rate limited sun-nadir steering, declination-limited sun-nadir steering, and the method disclosed in U.S. Pat. No. 5,794,891 issued to Polle et al. In sun-target steering, the yaw axis is pointed at a target other than nadir, such as a ground-fixed point. In rate-limited sun-nadir steering, the spacecraft yaw rate is limited, and the yaw rate is allowed to lag or lead the conventional sun-nadir profile to accommodate this yaw rate-limited configuration. In declination-limited steering, when the sunline is inconveniently close to the orbital plane, the spacecraft body is held orbit normal, and the solar array is pointed by a pitch gimbal.
On many spacecraft it is desirable to employ concentrator solar arrays, which provide more power per solar cell, thereby giving more power on a per unit cost basis. These concentrator arrays use mirrors or lenses that focus the sun's rays on small, high-temperature photovoltaic cells. However, these concentrator arrays typically must be pointed at the sun with a very high degree of accuracy in order to generate enough power to meet the bus requirements. The required pointing accuracy typically renders concentrator arrays unsuitable for use on satellites programmed for sun-nadir steering due to the sun tracking pointing error inherent when effectuating the noon turn (simply put, the satellite must "flip" at solar noon and solar midnight). The closer the sun lies to the orbit plane, the faster this "flip" must be done to point the arrays exactly at the sun.
Ideally, a spacecraft programmed for sun-nadir steering would effectuate the noon turn instantaneously when the sun is in the orbital plane. However, in practice the noon turn is both rate and acceleration limited, and thus there will always be sun tracking pointing error during portions of the turn in this case. Nevertheless, it would be highly desirable to have a spacecraft programmed for sun-nadir steering that is equipped with concentrator arrays, and which minimizes sun-tracking pointing error during portions of the noon-turn.
The solar arrays for satellites programmed for sun-nadir steering are conventionally sized to account for the fact that the solar arrays will experience power loss due to sun tracking pointing error during portions of the noon turn. This power loss is a especially problematic on high power satellites or on satellites in a low earth orbit having high orbital rates, which sometimes have orbital periods of as low as ninety minutes. The power loss when the solar arrays are not perpendicular to the sunlight is not significant for non-concentrated solar arrays because the power loss due to the non-perpendicularity goes roughly with the cosine of the angle away from perpendicularity. Therefore, an error angle of 25 degrees still allows for cosine(25 degrees)=0.906 of the power--over 90%.
For concentrator panels, however, the reduction in power with error angle is typically linear. Consequently, conventional sun-nadir satellites with concentrator arrays would require relatively large solar arrays and additional batteries, all of which increases weight, in order to account for the resulting loss in sunlight exposure due to sun tracking pointing error. Unfortunately, the extra weight increases the rotational moment of inertia of the spacecraft, which in turn necessitates the use of larger reaction wheels to perform the noon turn. The larger reaction wheels in turn increase the spacecraft weight even more. As the weight increases, the achievable slew rate is reduced, which negatively impacts the sun tracking pointing error during the noon turn.
It is known to point a solar array accurately at the sun by means of a two-axis gimbal between the solar array and the spacecraft body. Such a system is described, for example, in Fisher et al., "Magnetic Momentum Bias Control With Two-Gimballed Appendages", Paper No. AAS 95-005, at page 72, which can be found in Volume 88 of Advances in the Astronautical Sciences, published for the American Astronautical Society. As discussed therein, the body is held fixed with respect to nadir and the orbit, the inner gimbal rotates at orbit rate, and the outer gimbal tracks out the angle between the orbit and the sunline. This approach has many disadvantages. First, the outer gimbal deflections required can be very large (the Fisher article shows 90 degrees) and can stay that way for many orbits. Such large and persistent gimbal travel sweeps the array through a spacecraft body field of view, potentially intruding into the fields of view of sensors, the payload field of view, thermal radiators, and even into thruster plumes. It also creates large variations in the spacecraft inertia matrix and can create severe gravity gradient torques in low orbits. Furthermore, the benefits of sun-nadir steering in limiting the momentum buildup from solar torques, and of limiting the directions that sunlight can intrude on radiators, payloads, etc., are lost. Thus, it would be desirable to avoid, minimize, or even eliminate one or more of the above-cited problems.